Invertible unital bimodules over rings with local units, and related exact sequences of groups, II

Journal of Algebra 370 (2012) 266–296

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Journal of Algebra www.elsevier.com/locate/jalgebra

Invertible unital bimodules over rings with local units, and related exact sequences of groups, II ✩ L. El Kaoutit a , J. Gómez-Torrecillas b,∗ a b

Departamento de Álgebra, Facultad de Educación y Humanidades, Universidad de Granada, El Greco No. 10, E-51002 Ceuta, Spain Departamento de Álgebra, Facultad de Ciencias, Universidad de Granada, E18071 Granada, Spain

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 21 March 2011 Available online xxxx Communicated by Michel Van den Bergh Keywords: Rings with local units Unital bimodules Picard groups Brauer groups

Let R be a ring with a set of local units, and a homomorphism of groups Θ : G → Pic( R ) to the Picard group of R. We study under which conditions Θ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension R ⊆ S with the same set of local units and assuming that Θ is induced by a homomorphism of groups G → Inv R ( S ) to the group of all invertible R-sub-bimodules of S, then we construct an analogue of the Chase–Harrison– Rosenberg seven terms exact sequence of groups attached to the triple ( R ⊆ S , Θ), which involves the first, the second and the third cohomology groups of G with coefficients in the group of all R-bilinear automorphisms of R. Our approach generalizes the works by Kanzaki and Miyashita in the unital case. © 2012 Elsevier Inc. All rights reserved.

Introduction For a Galois extension R /k of commutative rings with identity and finite Galois group G, Chase, Harrison and Rosenberg, gave in [5, Corollary 5.5] (see also [6]) the following seven terms exact sequence of groups

1





H 1 G, U (R)



Pick (k)



H 1 G , Pic R ( R )

✩

*



Pic R ( R )G





H 2 G, U (R)

B ( R /k)



H 3 G, U (R) ,

Research supported by the grant MTM2010-20940-C02-01 from the Ministerio de Ciencia e Innovación and from FEDER. Corresponding author. E-mail addresses: [email protected] (L. El Kaoutit), [email protected] (J. Gómez-Torrecillas).

0021-8693/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2012.07.035

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267

where for a ring A with identity, Pic A ( A ) = {[ P ] ∈ Pic( A ) | ap = pa, ∀a ∈ A , p ∈ P } is a sub-group of the Picard group Pic( A ), and U ( A ) the group of units. The group B ( R /k) is the Brauer group of Azumaya k-algebras split by R, and the other terms are the usual cohomology groups of G with coefficients in abelian groups. As one can realize, the case of finite Galois extension of fields reduces to fundamental theorems in Galois cohomology of fields extensions, namely, the Hilbert’s 90 Theorem, and the isomorphism of the Brauer group of a field with the second cohomology group. The construction of this sequence of groups, as was given in [5,6], is ultimately based on the consideration of a certain spectral sequence already introduced by Grothendieck, and the generalized Amitsur cohomology. The same sequence was also constructed after that by Kanzaki [13, Theorem, p. 187], using only elementary methods that employ the novel notion of generalized crossed products. This notion has been the key success which allowed Miyashita to extend in [14, Theorem 2.12] the above framework to the context of a noncommutative unital ring extension R ⊆ S with a homomorphism of groups to the group of all invertible sub-R-bimodules of S. In this setting the seven terms exact sequence takes of course a slightly different form, although the first, the fourth and the last terms remain the cohomology groups of the acting group with coefficients in the group of units of the base ring. The commutative Galois case is then recovered by first considering a tower k ⊆ R ⊆ S, where k ⊆ R is a finite Galois extension and S is a fixed crossed product constructed from the Galois group; secondly by considering the homomorphism of groups which sends any element in the Galois group to it corresponding homogeneous component of S (components which belong to the group of all invertible sub-R-bimodules of S). In this paper we construct an analogue of the above seven terms exact sequence in the context of rings with local units (see the definition below). The most common class of examples are the rings with enough orthogonal idempotents or Gabriel’s rings which are defined from small additive categories. It is noteworthy that up to our knowledge there is no satisfactory notion in the literature of morphism of rings with local units that could be, for instance, extracted from a given additive functor between additive small categories. Here we restrict ourselves to the naive case of extensions of the form R ⊆ S which have the same set of local units. This at least includes the situation where two small additive categories have the same set of objects and differ only in the size of the sets of morphisms, as it frequently happens in the theory of localization in abelian categories. Our methods are inspired from Miyashita’s work [14] and use the results of our earlier work [9]. The present paper is in fact a continuation of [9]. It is worth noticing that, although the steps for constructing the seven terms sequence in our context are slightly parallel to [14], this construction is not an easy task since it has its own challenges and difficulties. The paper is organized as follows. In Section 1 we give some preliminary results on similar and invertible unital bimodules. Most of results on similar unital bimodules are in fact the non-unital version of [11,12], except perhaps the construction of the twisting natural transformations and its weak associativity (Proposition 1.3 and Lemma 1.4). The generalized crossed product is introduced in Section 2, where we also give useful constructions on the normalized 2- and 3-cocycles with coefficient in the unit group of the base ring. Section 3 is devoted to the construction of an abelian group whose elements consist of isomorphic classes of generalized crossed products. We show that it contains a sub-group which is isomorphic to the 2-cohomology group (Proposition 3.2). Our main result is contained in Section 4, which contains a number of exact sequences of groups that culminate in the seven terms exact sequence that generalizes Chase–Harrison–Rosenberg’s one (Theorem 4.12). Notations and basic notions In this paper ring means an associative and not necessarily unital ring. We denote by Z( R ) (resp. Z(G )) the center of a ring R (resp. of a group G), and if R is unital we will denote by U ( R ) the unit group of R, that is, the set of all invertible elements of R. Let R be a ring and E ⊂ R a set of idempotent elements; R is said to be a ring with set of local units E, provided that for every finite subset {r1 , . . . , rn } ⊂ R, there exists e ∈ E such that

er i = r i e = r i ,

for every i = 1, . . . , n.

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In this way, given a finite set {r1 , . . . , rn } of elements in R, we denote

Unit{r1 , . . . , rn } = {e ∈ E | er i = r i e = r i , for every i = 1, 2, . . . , n}. Observe that our definition differs from that in [1, Definition 1.1], since we do not assume that the idempotents of E commute. In fact, our rings generalize those of [1] since, when the idempotents are commuting, it is enough to require that for each element r ∈ R there exists e ∈ E such that er = re = r (see [1, Lemma 1.2]). The Rees matrix rings considered in [2], are for instance, rings with a set of local units with noncommuting idempotents. Let R be a ring with a set of local units E. A right R-module X is said to be unital provided one of the following equivalent conditions holds

∼ (i) X ⊗ R R  = X via the right R-action on X , (ii) X R := { finite xi r i | r i ∈ R , xi ∈ X } = X , (iii) for every element x ∈ X , there exists an element e ∈ E such that xe = x. Left unital R-modules are analogously defined. Obviously any right R-module X contains X R as the largest right unital R-submodule. Let R and S be rings with a fixed set of local units, respectively, E and E . In the sequel, an extension of rings φ : ( R , E) → ( S , E ) with the same set of local units, stand for an additive and multiplicative map φ which satisfies φ(E) = E . Note that since R and S have the same set of local units, any right (resp. left) unital S-module can be considered as right (resp. left) unital R-module by restricting scalars. Furthermore, for any right S-module X , we have the equality X R = X S. 1. Similar and invertible unital bimodules Let R be a ring with a set of local units E. In the first part of this section we give some preliminary results on unital R-bimodules, which we will use frequently in the sequel. Most of them were already formulated by K. Hirata in [11] and [12] when R is unital. For the convenience of the reader we will give in our case complete proofs. The second part provides a relation between the automorphism group of an invertible unital R-bimodule and the automorphisms of the base ring R. This result will be also used in the forthcoming sections. A unital R-bimodule is an R-bimodule which is left and right unital. In the sequel, the expression R-bilinear map means homomorphism of R-bimodules. Note that if M is a unital R-bimodule, then for every element m ∈ M, there exists a two sided unit e for m. That is, there exists e ∈ E such that m = me = em, see [7, p. 733]. If several R-bimodules are handled, say M , N , P with a finite number of elements {mi , ni , p i }i , mi ∈ M, ni ∈ N, p i ∈ P , then we denote by Unit{mi , ni , p i } the set of common two sided units of the mi ’s, ni ’s and the p i ’s. This means, that e ∈ Unit{mi , ni , p i } ⊆ E if and only if

emi = mi = mi e ,

eni = ni = ni e ,

ep i = p i = p i e ,

for every i = 1, . . . , n.

Given a unital R-bimodule M, we denote its invariant sub-bimodule, that is, the set of all Rbilinear maps from R to M, by

M R := Hom R - R ( R , M ). It is canonically a module over the commutative ring of all R-bimodule endomorphisms Z := End R - R ( R ). The Z -action is given as follows: for every z ∈ Z and f ∈ M R , we have

z. f : R R R −→ R M R









r −→ f z(r ) = z(e ) f (r ) = f (r ) z(e ), where e ∈ Unit{r } .

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269

⊕

Lemma 1.1. Let R be a ring with local units and M a unital R-bimodule such that M isomorphic as an R-bimodule to a direct summand of direct sum of n copies of R. Then

R (n) , that is, M is

(i) M R is a finitely generated and projective Z -module; (ii) there is an isomorphism of R-bimodules

M∼ = R ⊗Z M R , where the R-bimodule structure of the right hand term is induced by that of R. That is,



r (s ⊗Z f )t = (rst ) ⊗Z f



s ⊗Z f ∈ R ⊗Z M R , r , t ∈ R .

Proof. Let us consider the following canonical R-bimodule homomorphisms:

ϕi : M

ι

R (n)

πi

ιi

ψi : R

R,

R (n)

π

M,

i = 1, . . . , n ,

where πi and ιi are the canonical projections and injections. (i) It is clear that {(ψi , ψi∗ )}i ∈ M R × HomZ ( M R , Z ) is a finite dual Z -basis for M R , where each ψi∗ is defined by the right composition with ϕi , that is, ψi∗ ( f ) = ϕi ◦ f , for every f ∈ M R . (ii) We know that there is a well defined R-bilinear map η

R ⊗Z M R r ⊗Z f

M f (r ).

We can easily show that

M m is actually the inverse map of

η −1

n i

R ⊗Z M R

ϕi (m) ⊗Z ψi ,

η. 2

We will follow Miyashita’s notation concerning this class of R-bimodules. That is, if we have an R-bimodule M which is a direct summand as a bimodule of finitely many copies of another R-bimodule N, we shall denote this situation by M | N. It is clear that this relation is reflexive and transitive. Furthermore, it is compatible with the tensor product over R, in the sense that we have M ⊗ R Q | N ⊗ R Q and Q ⊗ R M | Q ⊗ R N, for every unital R-bimodule Q , whenever M | N. In this way, we set

M∼N

if and only if

M | N and N | M ,

(1)

and call M is similar to N. This in fact defines an equivalence relation which is, in the above sense, compatible with the tensor product over R. Lemma 1.2. Let R be a ring with local units and M, N are unital R-bimodules such that M | R and N | R as bimodules. Then

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(i) for every finitely generated and projective Z -module P , we have an isomorphism of Z -modules

( R ⊗Z P ) R ∼ = P; (ii) there is a natural (on both components) Z -linear isomorphism

(M ⊗R N )R ∼ = M R ⊗Z N R . Proof. (i) Let {( p i , p ∗i )}1i n be a dual Z -basis for P . Given an element f ∈ ( R ⊗Z P ) R , we define a finite family of elements in Z :





z f ,i := R ⊗Z p ∗i ◦ f : R −→ R ⊗Z P −→ R ⊗Z Z ∼ = R. Now, an easy computation shows that the following maps are mutually inverse

P p

( R ⊗Z P ) R

( R ⊗Z P ) R

[r −→ r ⊗Z p ],

f

P

n i

z f ,i p i ,

which gives the stated isomorphism. (ii) By Lemma 1.1(ii) we know that

M∼ = R ⊗Z M R

and

N∼ = R ⊗Z N R .

Therefore, we have





M ⊗R N ∼ = R ⊗Z M R ⊗Z N R . Since by Lemma 1.1(i), M R ⊗Z N R is a finitely generated and projective Z -module, we obtain the desired isomorphism by applying item (i) just proved above. 2 Keep the notations of the first paragraph in the proof of Lemma 1.1. Proposition 1.3. Let R be a ring with local units and M, N are unital R-bimodules such that M | R and N | R as bimodules. Then there is a natural R-bilinear isomorphism M ⊗ R N ∼ = N ⊗ R M given explicitly by ∼ =

TM ,N : M ⊗ R N

n

x ⊗R y

i

N ⊗R M

ϕi (x) y ⊗ R ψi (e),

where e ∈ Unit{x, y }. Proof. By Lemmata 1.1 and 1.2, we have the following chain of R-bilinear isomorphisms

M ⊗R N

∼ = ∼ =

R ⊗Z M R ⊗Z N R

∼ =

R ⊗Z N R ⊗Z M R

∼ =



N ⊗ R R ⊗Z M R



N ⊗R M .

Writing down explicitly the composition of these isomorphisms, we obtain the stated one.

2

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The natural transformation T−,− is associative in the following sense. Lemma 1.4. Let R be ring with a local units. Consider X , Y , P , Q , U , V unital R-bimodules which satisfy Q | R, (Y ⊗ R V )| R, ( X ⊗ R U )| R and ( P ⊗ R Y )| R, all as bimodules. Then the following diagram is commutative: X ⊗ U ⊗ P ⊗ TY ⊗ V , Q

X ⊗R U ⊗R P ⊗R Y ⊗R V ⊗R Q

X ⊗R U ⊗R P ⊗R Q ⊗R Y ⊗R V

TX⊗ U ,P ⊗ Y ⊗ V ⊗ Q

TX⊗ U ,P ⊗ Q ⊗ Y ⊗ V

P ⊗R Y ⊗R X ⊗R U ⊗R V ⊗R Q

Proof. Straightforward.

P ⊗R Q ⊗R Y ⊗R X ⊗R U ⊗R V .

P ⊗ TY ⊗ X ⊗ U ⊗ V , Q

2

Recall form [9, p. 227] that a unital R-bimodule X is said to be invertible if there exists another unital R-bimodule Y with two R-bilinear isomorphisms

X ⊗R Y

l

r

R

∼ =

∼ =

Y ⊗R X .

As was shown in [9], one can choose the isomorphisms l, r such that

X ⊗R r = l ⊗R X ,

Y ⊗R l = r ⊗R Y .

In others words, such that (l, r−1 ) is a Morita context from R to R. In this way, Y is isomorphic as an R-bimodule to the right unital part X ∗ R of the right dual R-module X ∗ = Hom− R ( X , R ) of X . This isomorphism is explicitly given by ∼ =

Y− −→ X ∗ R







y −→ x −→ r( y ⊗ x) .

It is worth mentioning that X is not necessarily finitely generated as right unital R-module; but a direct limit of finitely generated and projective right unital R-modules. Given e ∈ E consider its decomposition with respect to the invertible unital R-bimodule X , that is,

e=



 xe y e

i.e. e =

(e )



l(xe ⊗ R y e ) .

(e )

It is clear that the xe ’s and the y e ’s can be chosen such that

exe = xe ,

and

ye = ye e,

which means that e is a right unit for the y e ’s and left unit for the xe ’s. Remark 1.5. It is noteworthy that a two sided unit of the set of elements {xe , y e } does not need in general to coincide with e.  However, any two sided unit e 1 ∈ Unit{xe , y e } is also a unit for e. In other words, the elements e and (e ) xe ey e belong to the same unital ring e Re, but they are not necessarily equal. This makes a difference in technical difficulties compared with the case of unital rings.

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The following lemma gives explicitly the relation between the automorphisms group of unital invertible R-bimodule and the unit group of the commutative unital ring Z = End R - R ( R ). Lemma 1.6. Let R be a ring with local units and X an invertible unital R-bimodule. Then there is an isomorphism of groups, from the group of R-bilinear automorphisms of X , Aut R - R ( X ) to the unit group of Z , defined by

(−)

Aut R - R ( X )

Aut R - R ( R ) = U (Z )

(2)

 σ

σ where for each r ∈ R with unit e ∈ Unit{r }, we have

 σ (r ) =



σ (xe ) y e r =

(e )



r σ (xe ) y e .

(e )

In particular, for every element t ∈ X and σ ∈ Aut R - R ( X ), we have

σ (t) =  σ (e)t, whenever e ∈ Unit{t}. Proof. Follows directly from [4, Lemma 2.1].

2

The Picard group of R is denoted by Pic( R ). It consists of all isomorphisms classes of invertible unital R-bimodules, with multiplication induced by the tensor product. As in the unital case [10, Theorem 2(i)], there is a canonical homomorphism of groups α : Pic( R ) → Aut(U (Z )). This homomorphism is explicitly given as follows. Given [ X ] ∈ Pic( R ) and u ∈ U (Z ), we consider the R-bilinear automorphism of X defined by

σu : X −→ X We clearly have



 t −→ tu (e ), t ∈ X , and e ∈ Unit{t} .

u ◦ σ v , for u , v ∈ U (Z ). It follows from Lemma 1.6 that σuv = σu ◦ σ v , and so σ

uv = σ

σu (r t) = σu (r )t, for every t ∈ X , r ∈ R .

(3)

This equation implies that the element σu ∈ U (Z ) is independent from the choice of the representative X of the class [ X ] ∈ Pic( R ). We have thus a well defined map





α : Pic( R ) −→ Aut U (Z ) , [ X ] −→ α [ X ] (u ) = σu which is a homomorphism of groups by (3). This is exactly the homomorphism induced by the map of [4, p. 135]. We have the formula:

α [ X ] (u )(r ) = σu (r ) =



 rxe u (e 1 ) y e

i.e.

(e )

where e ∈ Unit{r }, e 1 ∈ Unit{xe , y e }, and as before e =

   r l xe ⊗ R u (e 1 ) y e , (e )



(e ) xe y e .

α

(4)

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273

2. Generalized crossed products with local units Let G be any group with neutral element 1. Consider a ring R with a set of local units E, and a group homomorphism Θ : G → Pic( R ). This is equivalent to give sets of invertible R-bimodules and isomorphisms of bimodules

{Θx : x ∈ G },



Θ ∼ = Fx, y : Θx ⊗ R Θ y −− → Θxy , x, y ∈ G , 

where Θ(x) = [Θx ] for every x ∈ G . Consider the product on the direct sum x∈G Θx determined by the maps FxΘ, y . There is no reason to expect that this product, after the choice of the representatives Θx , will be associative. In fact, it becomes an associative multiplication if and only if the diagram

Θx ⊗ R Θ y ⊗ R Θ z

Θ ⊗Θ Fxy z

Θxy ⊗ R Θz

Θ Θx ⊗ F yz

(5)

Θ Fxy ,z

FxΘ, yz

Θx ⊗ R Θ yz

Θxyz

commutes for every x, y , z ∈ G . Even in this case, it is not clear whether this multiplication has a set of local units. On the other hand, we know that there is an isomorphism of R-bimodules ι : R → Θ1 , so a natural candidate for such a set should be E = ι(E). In fact, this is a set of local units if and only if, for every x ∈ G , the following diagrams are commutative 

Θx ⊗ R R Θx ⊗ ι

Θx ⊗ R Θ1

Θx , FxΘ,1



R ⊗ R Θx

Θx . (6)

ι⊗Θx

Θ1 ⊗ R Θx

F1Θ,x

The previous discussion suggest then the following definition. Definition 2.1. A factor map for the morphism Θ : G → Pic( R ) consists of sets of invertible R-bimodules and isomorphisms of bimodules

{Θx : x ∈ G },



Θ ∼ = Fx, y : Θx ⊗ R Θ y −− → Θxy , x, y ∈ G ,

where Θ(x) = [Θx ] for every x ∈ G , and one isomorphism of R-bimodules diagrams (5) and (6) are commutative.

ι : R → Θ1 such that the

Now we come back to our initial homomorphism of groups Θ : G → Pic( R ) and its associated family of isomorphisms {FxΘ, y }x, y ,∈G . Our next aim is to find conditions under which we can extract from these data a factor map for R and G . First of all we should assume that the diagrams of (6) are commutative. We can define using the homomorphism of groups α : Pic( R ) → Aut(U (Z )) given by Eq. (4), a left G -action on the group of units U (Z ), where Z = End R - R ( R ). Written down explicitly the composition α ◦Θ : G → Pic( R ) → Aut(U (Z )) the outcome action is described as follows. Given  a unit e ∈ E, its decomposition with respect to the invertible R-bimodule Θx is written as e = (e) xe xe ∈

ι−1 (Θ1 ) = R. Using formula (4), the left G -action on U (Z ) is then defined by

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G × U (Z )

U (Z )

 x

(x, u )

u = r −→



 rxe u (e 1 )xe ,

(7)

(e )

where e ∈ Unit{r } and e 1 ∈ Unit{xe , xe }. Recall that in this case e 1 ∈ Unit{e }, and so e 1 ∈ Unit{r }. n The abelian group of all n-cocycles with respect to this action is denoted by Z Θ (G , U (Z )); while n by B Θ (G , U (Z )) we denote the n-coboundary sub-group. The n-cohomology group, is then denoted by H nΘ (G , U (Z )). Remark 2.2. To each homomorphism of groups Θ : G → Pic( R ), it corresponds as before a G -group ∗ (G , U (Z )). It seems that this corstructure on U (Z ), and so it leads to the cohomology groups H Θ respondence has not been systematically studied, even in the case of unital rings. However, as we will see, it is in fact the starting key point of any theory of generalized crossed products. Following the commutative and unital case, see Remark 2.13 below, here we will also extract our results from one fixed G -action, this is to say fixed cohomology groups. For instance, one could start with the cohomology which corresponds to the trivial homomorphism of groups, that is, I : G → Pic( R ) sending x → [ R ], of course here the cohomology is not at all trivial. Here is a non-trivial example. Example 2.3. (See [4].) Let R be a ring with E as a set of local units. Assume that there is a homomorphism of groups Φ : G → Aut( R ) to the group of ring automorphisms of R (we assume that Φ(x)(E) = E, for every x ∈ G ). Composing with the canonical homomorphism Aut( R ) → Pic( R ) which sends x → [ R x ] (i.e. the R-bimodule whose underlying left R-module is R R and its right module structure is r .s = rx(s)), we obtain a homomorphism Θ : G → Pic( R ). The corresponding family of maps {FxΘ, y }x, y∈G is given by

FxΘ, y : R x ⊗ R R y −→ R xy





r ⊗ R s −→ rx(s) .

Here of course the associativity and unitary properties of the multiplication of R, both imply that {FxΘ, y }x, y∈G is actually a factor map for R and G . A routine computation shows now that the coho∗ (G , U (Z )) which corresponds to this Θ is exactly the one considered in [4, p. 135], since mology H Θ the G -action (7) coincides for this case with that given in [4]. Lemma 2.4. Keep the above G -action on U (Z ). Fix an element x ∈ G . (1) For every t ∈ Θx and u ∈ U (Z ), we have

tu (e ) = x u (e )t,

where e ∈ Unit{t}.

(8)

(2) Given similar bimodules M ∼ Θx ; for every m ∈ M and u ∈ U (Z ), we have

mu (e ) = x u (e )m,

with e ∈ Unit{m}.

(9)

Proof. (1) This is Eq. (3). (n) (2) The identity (8) clearly lifts to finite direct sums Θx , and, hence, for sub-bimodules M (n) of Θx . 2

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Proposition 2.5. Let R be a ring with a set of local units E and G any group. Assume that there is a homomorphism of groups Θ : G → Pic( R ) whose family of maps {FxΘ, y }x, y ∈G satisfy the commutativity of diagrams (6). Denote by αx, y ,z : Θxyz → Θxyz the isomorphisms that satisfy









Θ Θ αx, y,z ◦ FxΘ, yz ◦ Θx ⊗ R F yΘ,z = Fxy , z ◦ Fx, y ⊗ R Θz .

(10)

3   Then α x, y , z defines a normalized element of Z Θ (G , U (Z )), where α x, y , z is the image of αx, y , z under the isomorphism of groups stated in Lemma 1.6. Furthermore, if α −,−,− is cohomologically trivial, that is, there is a normalized map σ−,− : G × G → U (Z ) such that

−1  αx, y,z = σx, yz x σ y,z σx−, y1 σxy ,z ,

for every x, y , z ∈ G , then there is a factor map {F Θ x, y }x. y ∈G for Θ , defined by

FΘ x, y : Θx ⊗ R Θ y

Θxy

ux ⊗ R u y

σx, y (e)FxΘ, y (ux ⊗ R u y ),

where e ∈ Unit{ux , u y }. Proof. Consider x, y , z, t ∈ G , and ux ∈ Θx , u y ∈ Θ y , uz ∈ Θz and ut ∈ Θt , denote by ux u y the image of ux ⊗ R u y under FxΘ, y . There are several isomorphisms from Θx ⊗ R Θ y ⊗ R Θz ⊗ R Θt to Θxyzt which are Θ . So it will be convenient to adopt some notations. In order to do this, we defined by the family F−,− rewrite the stated equation (10) satisfied by the α−,−,− ’s, as follows









αx, y,z ux (u y uz ) = (ux u y )uz . Using Lemma 1.6, this is equivalent to

    βx, y ,z (e ) ux (u y uz ) = (ux u y )uz , where e ∈ Unit{ux , u y , uz } and β−,−,− := α −,−,− is the image of αx, y , z under the homomorphism of groups defined in Lemma 1.6. In this way, if we fix a unit e ∈ Unit{ux , u y , uz , ut }, then we have



   (ux u y )uz ut = βx, y ,z (e ) ux (u y uz ) ut    = βx, y ,z (e )βx, yz,t (e ) ux (u y uz )ut     = βx, y ,z (e )βx, yz,t (e ) ux β y ,z,t (e ) u y (uz ut )    = βx, y ,z (e )βx, yz,t (e ) ux β y ,z,t (e ) u y (uz ut )    (8 ) = βx, y ,z (e )βx, yz,t (e ) x β y ,z,t (e )ux u y (uz ut )    = βx, y ,z (e )βx, yz,t (e )x β y ,z,t (e ) ux u y (uz ut )   = βx, y ,z (e )βx, yz,t (e )x β y ,z,t (e )βx−, 1y ,zt (e ) (ux u y )(uz ut )   −1 = βx, y ,z (e )βx, yz,t (e )x β y ,z,t (e )βx−, 1y ,zt (e )βxy , z,t (e ) (ux u y )u z ut .

Given h ∈ E, using the above notations, we can consider the following units

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h=



xh xh ,

h1 ∈ Unit{xh , xh };

(h)

h1 =



y h1 y h1 ,

h2 ∈ Unit{ y h1 , y h1 };

(h1 )

h2 =



zh 2 z h 2 ,

h3 ∈ Unit{ zh2 , zh2 };

t h3 t h3 ,

h4 ∈ Unit{th3 , t h3 }.

(h2 )

h3 =

 (h3 )

It is clear that h4 is a unit for all the handled elements, in particular h4 ∈ Unit{xh , y h1 , zh2 , th3 }. According to the equality showed previously, for every set {xh , y h1 , zh2 , th3 }, we have



   −1 (xh yh1 ) zh2 th3 = βx, y ,z (h4 )βx, yz,t (h4 )x β y ,z,t (h4 )βx−, 1y ,zt (h4 )βxy , z,t (h 4 ) (xh y h1 ) zh2 t h3 .

Using the map FtΘ,t −1 , and summing up, we get



   −1 (xh yh1 ) zh2 th3 t h3 = βx, y ,z (h4 )βx, yz,t (h4 )x β y ,z,t (h4 )βx−, 1y ,zt (h4 )βxy , z,t (h 4 ) (xh y h1 ) zh2 t h3 t h3 ,

(h3 )

which means that −1 (xh yh1 ) zh2 = βx, y ,z (h4 )βx, yz,t (h4 )x β y ,z,t (h4 )βx−, 1y ,zt (h4 )βxy , z,t (h 4 )(xh y h1 ) zh2 , equality in Θxyz .

Repeating thrice the same process, we end up with −1 h = βx, y ,z (h4 )βx, yz,t (h4 )x β y ,z,t (h4 )βx−, 1y ,zt (h4 )βxy , z,t (h 4 )h −1 = βx, y ,z (h)βx, yz,t (h)x β y ,z,t (h)βx−, 1y ,zt (h)βxy , z,t (h ),

since h4 h = hh4 = h and the involved maps are R-bilinear. Therefore, for any unit h ∈ E, we have

βxy ,z,t ◦ βx, y ,zt (h) = x β y ,z,t ◦ βx, yz,t ◦ βx, y ,z (h). Since the β−,−,− ’s are R-bilinear, this equality implies the 3-cocycle condition, that is,

βxy ,z,t ◦ βx, y ,zt = x β y ,z,t ◦ βx, yz,t ◦ βx, y ,z ,

for every x, y , z, t ∈ G .

A routine computation, using Eq. (8), shows the last statement.

2

Corollary 2.6. Let R be a ring with local units and G any group, and fix a factor map {Θx }x∈G for R and G with its associated cohomology groups. Assume that there is a family of invertible unital R-bimodules {Ωx }x∈G such that Ωx ∼ Θx (they are similar), for every x ∈ G , with a family of R-bilinear isomorphisms

Fx, y : Ωx ⊗ R Ω y −→ Ωxy . For x, y , z ∈ G , we denote by αx, y ,z : Ωxyz → Ωxyz the resulting isomorphisms which satisfy

αx, y,z ◦ Fx, yz ◦ (Ωx ⊗ R F y,z ) = Fxy,z ◦ (Fx, y ⊗ R Ωz ).

(11)

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277

3   Then α x, y , z defines an element of Z Θ (G , U (Z )), where α x, y , z is the image of αx, y , z under the isomorphism of groups stated in Lemma 1.6.

Proof. This is a direct consequence of Proposition 2.5 and Lemma 2.4(2).

2

We need to clarify the situation when two different classes of representatives are chosen. Proposition 2.7. Let R be a ring with a set of local units E and G any group. Consider two factor maps ∼ = {FxΘ, y }x, y∈G and {FxΓ, y }x, y∈G such that, for every x ∈ G , there exists an R-bilinear isomorphism ax : Γx − → Θx . Denote by τx, y , where x, y ∈ G , the following R-bilinear isomorphism

τx, y : Θxy

FxΘ, y −1

Θx ⊗ R Θ y

∼ =

Γx ⊗ R Γ y

FxΓ, y

Γxy

∼ =

Θxy .

That is,

τx, y ◦ FxΘ, y ◦ (ax ⊗ R a y ) = axy ◦ FxΓ, y . 2

Then τ

x, y defines a normalized element of Z Θ (G , U (Z )), where τ x, y is the image of phism of groups stated in Lemma 1.6.

τx, y under the isomor-

Proof. Fix a set of elements x, y , z ∈ G . Assume that we have x

τ

τx, yz (h) = τ

τxy,z (h), for every unit h ∈ E. y , z (h ) x, y (h )

(12)

So taking any element r ∈ R with unit f ∈ Unit{r }, we obtain x x



 τ x, yz ◦ τ y , z (r ) = τ x, yz ◦ τ y ,z ( f r )

= x τ

τx, yz (r ) y , z ( f ) = x τ

τx, yz ( f )r y , z ( f ) (12)

= τ

τxy,z ( f )r x, y ( f )  

= τ xy , z τ x, y ( f )r

= τ xy , z ◦ τ x, y (r ),  which shows that τ −,− is a 2-cocycle. In fact τ −,− is by definition a normalized 2-cocycle. Eq. (12) is fulfilled by following analogue steps as in the proof of Proposition 2.5. 2 Proposition 2.8. The hypothesis is that of Corollary 2.6. Assume further that there are two families of invertible unital R-bimodules {Ωx }x∈G and {Γx }x∈G as in Corollary 2.6 (i.e. Γx ∼ Θx ∼ Ωx ), together with families of Rbilinear isomorphisms

FxΩ, y : Ωx ⊗ R Ω y −→ Ωxy ,

FxΓ, y : Γx ⊗ R Γ y −→ Γxy .

Assume also that there are R-bilinear isomorphisms ax : Ωx → Γx , x ∈ G . Consider the associated 3-cocycles 3 Ω Γ β−,−,− , β−,−,− ∈ ZΘ (G , U (Z )) given by Proposition 2.5. Then,

− 1   Γ  Ω β−,−,− ◦ β−,−,− ∈ B 2Θ G , U (Z ) . 3 Ω Γ That is, they are cohomologous, or [β−,−,− ] = [β−,−,− ] in H Θ (G , U (Z )).

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Ω Γ Proof. Let us denote by α−,−,− and α−,−,− the R-bilinear isomorphisms satisfying Eq. (11), respectively, for {Ωx }x∈G and {Γx }x∈G . For any pair x, y ∈ G , we consider the R-bilinear isomorphism bxy : Ωxy → Ωxy defined by the following equation

axy ◦ bxy ◦ FxΩ, y = FxΓ, y ◦ (ax ⊗ R a y ).

(13)

We denote by γx, y , x, y ∈ G , the image of bxy under the homomorphism of groups defined in Lemma 1.6. Fix x, y , z ∈ G , and consider (tx , t y , tz ) ∈ Ωx × Ω y × Ωz and (ux , u y , uz ) ∈ Γx × Γ y × Γz with a common unit e ∈ Unit{tx , t y , tz , ux , u y , uz }. Using the notation of the proof of Proposition 2.5, we can write

βxΩ, y ,z (e )tx (t y tz ) = (tx t y )tz ,

(14)

βxΓ, y ,z (e )ux (u y uz ) = (ux u y )uz ,

(15)

γx, y (e)axy (tx t y ) = ax (tx )a y (t y ).

(16)

A routine computation as in the proof of Proposition 2.5 using Eqs. (14), (15) and (16), shows that, for every unit h ∈ E, we have



−1  −1 x −1 βxΩ, y ,z (h) βxΓ, y ,z (h) = γx, yz (h) γ y,z (h) γx, y (h)γxy,z (h).

This implies that

− 1 − 1   βxΓ, y ,z ◦ βxΩ, y ,z = γxy ,z ◦ γx, y ◦ x γ y ,z ◦ (γx, yz )−1 , which finishes the proof.

in U (Z ),

2

Now we are able to give the right definition of generalized crossed product. We hope it will result cleaner than the one considered in [13,14] for rings with identity. In the sequel, R denotes a ring with a set of local units E. Definition 2.9. Given a factor map as in Definition 2.1

 ∼ = FxΘ, y : Θx ⊗ R Θ y −− → Θxy , x, y ∈ G ,

with an  isomorphism of R-bimodules (Θ) = x∈G Θx with multiplication

ι : R → Θ1 , we define its associated generalized crossed product

θx θ y = FxΘ, y (θx ⊗ θ y ),

for θx ∈ Θx , θ y ∈ Θ y .

It follows that Θ1 is a subring of (Θ) and the map ι : R → Θ1 is a ring isomorphism. Therefore, ι(E) is a set of local units for Θ1 that serves as well as a set of local units for (Θ). Normally, we will identify R with Θ1 , and, thus, E will be a set of local units for the generalized crossed product (Θ). Obviously, (Θ) is a G -graded ring such that Θx Θ y = Θxy for every x, y ∈ G (thus, it could be referred to as a strongly graded ring after [15]). For instance, if we take Θ as in Example 2.3, then (Θ) is the skew group ring R ∗ G , see [4].

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279



Remark 2.10. A factor map as in Definition 2.1 determines a generalized crossed product Γ = x∈G Γx with ring isomorphism ι : R → Γ1 and multiplication given by γx γ y = FxΓ, y (γx ⊗ R γ y ) for γx ∈ Γx , γ y ∈ Γ y . In other words, generalized crossed products and factor maps are just two ways to define the same mathematical object. A generalized crossed product Γ gives clearly a group homomorphism



Γ : G −→ Pic( R )



x −→ [Γx ] .

Whether a general group homomorphism G → Pic( R ) gives some generalized crossed product is not so clear. However, as we have seen in Proposition 2.5, this is possible, if the underlying maps satisfy the commutativity of diagrams (6) and the associated 3-cocycle is trivial or at least cohomologically trivial. Definition 2.11. A morphism of generalized crossed products ((Θ), ν ), ((Γ ), ι) is a graded ring homomorphism f : (Θ) → (Γ ) such that f ◦ ν = ι. Equivalently, it consists of a set of homomorphisms of R-bimodules

{ f x : Θx −→ Γx : x ∈ G } such that all the diagrams

Θx ⊗ R Θ y

FxΘ, y

Θxy

fx⊗ f y

f xy

Γx ⊗ R Γ y

Γxy

FxΓ, y

commute, and f 1 ◦ ν = ι. Let R ⊆ S be an extension of rings with the same set of local units E. We denote as in [9] by Inv R ( S ) the group of all invertible unital R-sub-bimodules of S. An R-sub-bimodule X ⊆ S belongs to the group Inv R ( S ) if and only if there exists an R-sub-bimodule Y ⊆ S such that

XY = R = Y X, where the first and last terms are obviously defined by the multiplication of S. Clearly, there is a homomorphism of groups μ : Inv R ( S ) → Pic( R ). Remark 2.12. A generalized crossed product Γ of R with G gives in particular the ring extension ι : ( R , E ) → (Γ, E ) with local units. In this way, Γx is a unital R-sub-bimodule of Γ for every x ∈ G , and the isomorphism ι : R → Γ1 becomes R-bilinear. We will identify Γ1 with R and E with E . Let us denote by F Γ : Γ ⊗ R Γ → Γ the multiplication map of the ring Γ . It follows from [9, Lemma 1.1] that its restriction to Γx ⊗ R Γ y gives an R-bilinear isomorphism Γ :Γ ⊗ Γ Fxy x R y



Γx Γ y = Γxy

for every x, y ∈ G , since, obviously, Γx ∈ Inv R (Γ ) for every x ∈ G . Clearly, our generalized crossed product determines a factor map in the sense of Definition 2.1. On the other hand, the associated

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homomorphisms of groups Γ : G → Pic( R ) actually factors throughout the morphism have a commutative diagram of groups Γ

G

μ. That is, we

Pic( R ) (17) μ

Γ

Inv R (Γ ). As we have seen in Remark 2.12, one can start with a fixed ring extension R ⊆ S with the same set of local units E, and then work with the Γ ’s defined this time by Γ : G → Inv R ( S ) instead of the Γ ’s. In this case the situation becomes much more manageable, in the sense that each morphism Γ induces automatically a factor map and vice versa. Remark 2.13. As was shown in [13, Remarks 1, 2 and 3], the above notion generalizes the classical notion of crossed product in the commutative Galois setting. Specifically, given a commutative Galois extension k ⊆ R with a finite Galois group G , and consider the canonical homomorphism of groups Φ0 : G → Pick ( R ), where Pick ( R ) = {[ P ] ∈ Pic( R ) | kp = pk, ∀k ∈ k, p ∈ P }, which sends x → [ R x ] to the class of the R-bimodules R x whose underlying left R-module is R R and its right R-module structure was twisted by the automorphism x, see Example 2.3. In this case, the over  ring is chosen to be S = x∈G Φ0 (x) defined as the usual crossed product attached to Φ0 , which is not necessary trivial, that is, defined using any 2-cocycle. Thus, in the Galois theory of commutative rings, the homomorphism of groups which leads to the study of intermediate crossed products rings, is the obvious homomorphism of groups Θ which satisfies Φ0 = μ ◦ Θ , where μ is as in diagram (17). 3. An abelian group of generalized crossed products From now on, we fix a ring extension R ⊆ S with the same set of local units E. Let G be any group and assume given a morphism of groups

Θ : G −→ Inv R ( S ) (x −→ Θx ), where we have used the notation of Remark 2.12. The multiplication of S induces then a factor map

 FxΘ, y : Θx ⊗ R Θ y −→ Θxy : x, y ∈ G .

Clearly R = Θ1 , whence E is a set of local units for (Θ). We introduce in this section an abelian group C(Θ/ R ) whose elements are the isomorphism classes of generalized crossed products whose homogeneous components are similar as R-bimodules to that of (Θ), see Section 2 for definition. Our definition is inspired from that of Miyashita [14, §2], however the proofs presented here are slightly different. This group will be crucial for the construction of the seven terms exact sequence of groups in the next section. Let us consider thus the set C(Θ/ R ) of isomorphism classes of generalized crossed products [(Γ )] such that, for each x ∈ G , we have Γx ∼ Θx , that is they are similar as R-bimodules, see Section 1. Proposition 3.1. Consider the set of isomorphisms classes of generalized crossed products C(Θ/ R ) defined above. Then C(Θ/ R ) has the structure of an abelian group. Moreover the subset C0 (Θ/ R ) consisting of classes [(Λ)] such that for every x ∈ G , Λx ∼ = Θx as R-bimodules, is a sub-group of C(Θ/ R ).

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281

Proof. Given two classes [(Ω)], [(Γ )] ∈ C(Θ/ R ), their multiplication is defined by



  (Ω) (Γ ) =



 Ωx ⊗ R Θx−1 ⊗ R Γx .

x∈G

The factor maps of its representative generalized crossed product are given by the following composition

Ωx ⊗ R Θx−1 ⊗ R Γx ⊗ R Ω y ⊗ R Θ y −1 ⊗ R Γ y

Ω x ⊗ TΘ

x−1

⊗Γx ,Ω y ⊗ R Θ −1 ⊗Γ y y

Ωx ⊗ R Ω y ⊗ R Θ y −1 ⊗ R Θx−1 ⊗ R Γx ⊗ R Γ y

FxΩ, y ⊗ F Θ−1 −1 ⊗ FxΓ, y y ,x

FxΩΓ ,y

Ωxy ⊗ R Θ(xy )−1 ⊗ R Γxy , where TΘx−1 ⊗Γx ,Ω y ⊗ R Θ y−1 is the twist R-bilinear map defined in Proposition 1.3. The associativity of ΩΓ ’s is easily deduced using Lemma 1.4. This multiplication is commutative since for any two the F−,− classes [(Ω)], [(Γ )] ∈ C(Θ/ R ), we have a family of R-bilinear isomorphisms

∼ =

Γx ⊗ R Θx−1 ⊗ R Ωx

Γx ⊗ R Θx−1 ⊗ R Ωx ⊗ R Θx−1 ⊗ R Θx T⊗Θx

Ωx ⊗ R Θx−1 ⊗ R Γx ⊗ R Θx−1 ⊗ R Θx ∼ =

∼ =

Ωx ⊗ R Θx−1 ⊗ R Γx , which is easily shown to be compatible with both factors maps of Γ and Ω . Thus, it induces an isomorphism at the level of generalized crossed products. It is clear that the class of [(Θ)] is the unit of this multiplication. The inverse of any class [(Ω)] is given by



−1 (Ω) =

 x∈G

 Θx ⊗ R Ωx−1 ⊗ R Θx .

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Lastly, the fact that C0 (Θ/ R ) is a sub-group of C(Θ/ R ), can be immediately deduced from definitions. 2 Proposition 3.2. Consider the abelian group C0 (Θ/ R ) of Proposition 3.1. Then there is an isomorphism of groups

  2 C0 (Θ/ R ) ∼ G , U (Z ) . = HΘ Proof. The stated isomorphism is given by the following map:

     2 ζ : C0 (Θ/ R ) −→ H Θ G , U (Z ) (Λ) −→ [ τ−,− ] , where  τ−,− is the normalized 2-cocycle defined in Proposition 2.7 and [ τ−,− ] denotes its equivalence 2 class in H Θ (G , U (Z )). First we need to show that this map is in fact well defined. To do so, we take ∼ =

two isomorphic generalized crossed products χ : (Λ) −→ (Σ) which represent the same element in the sub-group C0 (Θ/ R ), with R-bilinear isomorphisms ax : Λx → Θx ← Σx : bx , for all x ∈ G . Then we check that the associated 2-cocycles are cohomologous. So let us denote these cocycles by  τ−,− −,− associated, respectively, to (Λ) and (Σ). We need to show that [ and γ τ−,− ] = [ γ−,− ]. Recall from Proposition 2.7 that we have

τx, y ◦ FxΘ, y ◦ (ax ⊗ R a y ) = axy ◦ FxΛ, y ,

γx, y ◦ FxΘ, y ◦ (bx ⊗ R b y ) = bxy ◦ FxΣ, y ,

Λ Σ for every x, y ∈ G , where F−,− and F−,− are, respectively, the factor maps relative to Λ and Σ . For each x ∈ G , we set the following R-bilinear isomorphism

βx : Θx

1 a− x

Λx

χx

bx

Σx

Θx ,

where χx is the x-homogeneous component of the isomorphism χ . Then it is easily seen, using the fact that χ is morphism of generalized crossed products, that, for each pair of elements x, y ∈ G , we have

βxy ◦ τx, y ◦ FxΘ, y = γx, y ◦ FxΘ, y ◦ (βx ⊗ R β y ).

(18)

We claim that

 τx, y (e)βxy (e) = γx, y (e)βx (e)x βy (e),

for all e ∈ E.

(19)



So let e ∈ E, and consider its decomposition e = (e) xe xe with respect to Θx , for a given x ∈ G . Let e 1 ∈ Unit{xe , xe }. Using Eq. (18), we get the following equality in S



βxy ◦ τx, y (xe y e1 ) y e1 xe =

(e ),(e 1 )







γx, y βx (xe )β y ( y e1 ) y e1 xe .

(e ),(e 1 )

Routine computations show that,

 

βxy τx, y (xe y e1 ) y e1 xe =  τx, y (e1 )βxy (e1 )e =  τx, y (e)βxy (e),

and

(e ),(e 1 )

(e ),(e 1 )





(8 )

γxy βx (xe )β y ( y e1 ) y e1 xe = γx, y (e1 )βx (e1 )x βy (e1 )e = γx, y (e)βx (e)x βy (e),

(20)

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283

which by (20) imply that

 τx, y (e)βxy (e) = γx, y (e)βx (e)x βy (e), which is the claimed equality. On the other hand we define

h:G

U (Z ) = Aut R - R ( R )

 h x : r −→

x



r βx (xe )xe

(e )

where e ∈ Unit{r }, and e =



(e ) xe xe is the decomposition of e in R

= Θx Θx−1 . It is clear form def-

x = h x , for ever x ∈ G . Taking an arbitrary element r ∈ R, we can show using Eq. (19) initions that β that

 τx, y ◦ h xy (r ) = γx, y ◦ x h y ◦ h x (r ). Therefore,  τx, y ◦ h xy = γx, y ◦ x h y ◦ h x , which means that  τ−,− and γ−,− are cohomologous. By the same way, we show that, if  τ−,− and γ−,− are cohomologous, then (Λ/ R ) and (Σ/ R ) are isomorphic as generalized crossed products, which means that the stated map ζ is injective. To show that this map is surjective, we start with a given normalized 2-cocycle σ−,− : G 2 → U (Z ), and consider the following R-bilinear automorphisms: for every x ∈ G

ϑx : Θx −→ Θx



 u −→ σx,x (e )u ,

where e ∈ Unit{u}. Now set Σx = Θx as an R-bimodule with the R-bilinear isomorphism ϑx : Σx → Θx previously defined. We define new factor maps relative to those Σx ’s, by

FxΣ, y : Σx ⊗ R Σ y −→ Σxy



 ux ⊗ R u y −→ σx, y (e )FxΘ, y (ux ⊗ R u y ) ,

Σ , while the norwhere e ∈ Unit{ux , u y }. The 2-cocycle condition on σ−,− gives the associativity of F−,− malized condition gives the unitary property. Thus, (Σ/ R ) is a generalized crossed product whose isomorphic class belongs by definition to the sub-group C0 (Θ/ R ). Lastly, by a routine computation, using the definition of the twist natural map given in Proposition 1.3, we show that the map ζ is a homomorphism of groups. 2

4. The Chase–Harrison–Rosenberg seven terms exact sequence In this section we show the analogue of Chase–Harrison–Rosenberg’s [5] seven terms exact sequence, generalizing by this the case of commutative Galois extensions with finite Galois group due of T. Kanzaki [13, Theorem, p. 187] and the noncommutative unital case treated by Y. Miyashita in [14, Theorem 2.12]. From now on we fix a ring extension R ⊆ S with a same set of local units E, together with a morphism of groups Θ : G → Inv R ( S ) and the associated generalized crossed product (Θ/ R ) having Θ F−,− as a factor map relative to Θ . We denote by Pic( R ) and Pic( S ), respectively, the Picard group of R and S. There is a canonical left G -action on Pic( R ) induced by the obvious homomorphism of groups Inv R ( S ) → Pic( R ) given explicitly by x

[ P ] = [Θx ⊗ R P ⊗ R Θx−1 ],

for every x ∈ G , and [ P ] ∈ Pic( R ).

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The corresponding G -invariant sub-group is denoted by Pic( R )G . We will consider the sub-group PicZ ( R ) whose elements are represented by Z -invariant R-bimodules, in the sense that

[ P ] ∈ PicZ ( R ) if and only if [ P ] ∈ Pic( R ),

and

  z(e ) pe = ep z e ,

∀p ∈ P ,

for every pair of units e , e ∈ E, and every element z ∈ Z . In this way, we set

PicZ ( R )G := PicZ ( R ) ∩ Pic( R )G . Recall from [9, Section 3] the group P( S / R ):

P( S / R ) = [ P ]

  [ X ]  [ P ] ∈ Pic( R ), [ X ] ∈ Pic( S ), and a map φ : R P R −→ R X R

[φ]

where at least one of the maps φ l : P ⊗ R S → X , or φ r : S ⊗ R P → X , canonically attached to φ , is an isomorphisms of bimodules. For more details on the structure group of this set we refer to [9, Section 3]. Obviously there is a homomorphism of groups

Ol : P( S / R ) −→ Pic( R )



[P ]

[φ]

  [X] −  → [P ] .

(21)

This group also inherits the above G -action. That is, there is a canonical left G -action on this group which is induced by Θ and given as follows:

x



[P ]

[φ]

  [ X ] = [Θx ⊗ R P ⊗ R Θx−1 ]

[ψ]

 [X] ,

where ψ := Θx ⊗ R φ ⊗ R Θx−1 (composed with the multiplication of S). The sub-group P( S / R )G of G -invariant elements of P( S / R ) has a slightly simpler description: Lemma 4.1. Keeping the above notations, we have

P( S / R )G = [ P ]

[φ]

  [ X ] ∈ P( S / R )  φ( P )Θx = Θx φ( P ), for every x ∈ G ,

where φ( P )Θx and Θx φ( P ) stand for the obvious subsets of the S-bimodule X , e.g.

φ( P )Θx =



   φ( p i )uix  p i ∈ P , uix ∈ Θx .

finite

Proof. Is based up on two main facts. The first one is that the R-bilinear map φ : P → X defining the given element [ P ]

[φ]

[ X ] ∈ P( S / R ), is always injective, see [9, Lemma 3.1]. The second

fact consists on the following equivalence: For a fixed x ∈ G , saying that φ( P )Θx = Θx φ( P ) in S X S is equivalent to say that there is an isomorphism f x : P ∼ = Θx ⊗ R P ⊗ R Θx−1 which completes the commutativity of the following diagram

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285

φ

P

X

fx

Θx ⊗ R P ⊗ R Θx−1

Θx ⊗φ⊗Θx−1

Θx ⊗ R X ⊗ R Θx−1

S ⊗R X ⊗R S

X.

ψ

The later means, by the definition of P( S / R ), that



[P ]

[φ]

  [ X ] = [Θx ⊗ P ⊗ Θx−1 ]

[ψ]

 [X] .

2

4.1. The first exact sequence Now we set

PZ ( S / R ) = [ P ]

[φ]

  [ X ] ∈ P( S / R )  [ P ] ∈ PicZ ( R ) ,

and

PZ ( S / R )G = PZ ( S / R ) ∩ P( S / R )G . On the other hand, we consider the group Aut R -ring ( S ) of all R-ring automorphisms of S (with same set of local units). It has the following sub-group





Aut R -ring ( S )(G ) = f ∈ Aut R -ring ( S )  f (Θx ) = Θx , for all x ∈ G . There are two interesting maps which will be used in the sequel: The first one is given by

E : Aut R -ring ( S ) −→ P( S / R )





f −→ [ R ]

[ι f ]

 [S f ] ,

where ι f is the inclusion R ⊂ S f , and S f is the S-bimodule induced from the automorphism f , that is, the left S-action is that of S S, while the right one has been altered by f , that is, we have a new left S-action

 

s.s = sf s ,

for all s, s ∈ S .

The second connects the group U (Z ) with Aut R -ring ( S ), and it is defined by

F : U (Z ) = Aut R - R ( R ) −→ Aut R -ring ( S )







σ −→ F(σ ) : s −→ σ −1 (e)sσ (e) ,

(22)

where e ∈ Unit{s}, s ∈ S. Proposition 4.2. Keeping the previous notations, there is a commutative diagram whose rows are exact sequences of groups

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U (Z ) = Aut R - R ( R )

F

E

Aut R -ring ( S )

Aut R -ring ( S )(G )

U (Z ) = Aut R - R ( R )

Ol

P( S / R )

PZ ( S / R )G

Pic( R )

PicZ ( R )G .

Proof. The exactness of the first row was proved in [9, Proposition 5.1]. The commutativity as well as the exactness of the second row, using routine computations, are immediately deduced from the definition of the involved maps. 2 The conditions under assumption, that is, the existence of Θ : G → Inv R ( S ) a homomorphism of groups with a given extension of rings R ⊆ S with the same set of local units E, are satisfied for the extension R ⊆ (Θ). Precisely, the map Θ factors thought out a homomorphism of groups Θ : G → Inv R ((Θ)) (x → Θx ), as R ⊆ (Θ) is an extension of rings with the same set of local units E. Thus, applying Proposition 4.2 to this extension, we obtain the first statement of the following corollary. Corollary 4.3. Consider the generalized crossed product  := (Θ) of S with G . Then there is an exact sequence of groups

Aut R -ring ()(G )

U (Z )

1

PZ (/ R )G

PicZ ( R )G .

In particular, we have the following exact sequence of groups

1





1 HΘ G , U (Z )

S1

PZ (/ R )G

S2

PicZ ( R )G .

Proof. We only need to check the particular statement. So let f ∈ Aut R -ring ()(G ) , this gives a family of R-bilinear isomorphisms f x : Θx → Θx , x ∈ G , which by Lemma 1.6 lead to a map

σ : G −→ U (Z ) (x −→ f x ), as Θx ∈ Inv R ( S ). An easy verification, using (8), shows that, for every unit e ∈ E, we have

 f xy (e ) =  f x (e )x f y (e ).

(23)

Therefore,

σxy = σx ◦ x σ y , for every x, y ∈ G . 1 That is, σ is a normalized 1-cocycle, i.e. σ ∈ Z Θ (G , U (Z )). 1 (G , U (Z )), we consider the R-bilinear isomorConversely, given a normalized 1-cocycle γ ∈ Z Θ phisms

g x : Θx −→ Θx Clearly, the direct sum g := g ∈ Aut R -ring ()(G ) .



x∈G





u −→ γx (e )u , where e ∈ Unit{u }, u ∈ Θx .

g x defines an automorphism of generalized crossed product. Hence,

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287

In conclusion we have constructed mutually inverse maps which in fact establish an isomorphism of groups





1 Aut R -ring ()(G ) ∼ G , U (Z ) . = ZΘ

(24)

Now let us consider an element u ∈ U (Z ) = Aut R - R ( R ), and its image F(u) under the map F defined in (22) using the extension R ⊆ . Then, for every element a ∈ Θx , we have

F(u)(a) = u−1 (e )au(e ), ( 8 ) −1

=u

e ∈ Unit{a}

(e )x u(e )a.

Therefore, u defines, under the isomorphism of (24), a 1-coboundary,

G −→ U (Z )





x −→ x u ◦ u−1 .

We have then constructed an isomorphism of groups





1 Aut R -ring ()(G ) /Aut R - R ( R ) ∼ G , U (Z ) . = HΘ

The exactness follows now from the first statement which is a particular case of Proposition 4.2.

2

4.2. The second exact sequence For any element [ P ] ∈ Pic( R ), we will denote by [ P −1 ] its inverse element. Let us consider the following groups





PicZ ( R )(G ) := [ P ] ∈ PicZ ( R )  P ⊗ R Θx ⊗ R P −1 ∼ Θx , for all x ∈ G . A routine computation shows that this is a sub-group of PicZ ( R ) which contains the sub-group of all G -invariant elements. That is, there is a homomorphism of groups PicZ ( R )G → PicZ ( R )(G ) . Given an element [ P ] ∈ PicZ ( R )(G ) , set Ωx := P ⊗ R Θx ⊗ R P −1 , for every x ∈ G . Consider now the family of R-bilinear isomorphisms

FxΩ, y : Ωx ⊗ R Ω y −→ Ωxy which is defined using the factor maps FxΘ, y and the isomorphism P ⊗ R P −1 ∼ = R. This is clearly a factor map relative to the homomorphism of groups Ω : G → Inv R ( S ) sending x → Ωx . This gives us a generalized crossed product (Ω), and in fact establishes a homomorphism of groups

L : PicZ ( R )(G ) −→ C(Θ/ R ), where the right hand group was defined in Section 3. The proof of the following lemma is left to the reader. Lemma 4.4. Keep the above notations. There is a commutative diagram of groups

PicZ ( R )G

L

PicZ ( R )(G )

L

C0 (Θ/ R )

C(Θ/ R ).

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Now we can show our second exact sequence. Proposition 4.5. Let (Θ) be a generalized crossed product of S with G . Then there is an exact sequence of groups

 G PZ (Θ)/ R

S2

S3

PicZ ( R )G

C0 (Θ/ R ),

where S2 and S3 are, respectively, the restrictions of the map Ol given in Eq. (21) attached to the extension R ⊆ (Θ), and of the map defined in Lemma 4.4. Proof. Let [ P ] ∈ PicZ ( R )G such that S3 ([ P ]) = [(Ω)] = [(Θ)], where for every x ∈ G , Ωx = P ⊗ R Θx ⊗ R P −1 . This means that we have an isomorphism (Ω) ∼ = (Θ) of generalized crossed products, and so a family of R-bilinear isomorphisms ιx : Θx ⊗ R P ∼ = P ⊗ R Θx , x ∈ G . We then obtain an Rbilinear isomorphism

ι=



ιx : (Θ) ⊗ R P −→ P ⊗ R (Θ),

x∈G

which in fact satisfies the conditions stated in [8, Eqs. (5.1) and (5.2), p. 161]. This implies that (Θ) ⊗ R P admits a structure of (Θ)-bimodule whose underlying left structure is given by that of (Θ) (Θ). In this way, the previous map ι becomes an isomorphism of ((Θ), R )-bimodules, and so we have



       (Θ) ⊗ R P ⊗(Θ) P −1 ⊗ R (Θ) ∼ = (Θ) ⊗ R P ⊗(Θ) (Θ) ⊗ R P −1 ∼ = (Θ) ⊗ R P ⊗ R P −1 ∼ = (Θ),

an isomorphism of (Θ)-bimodules. Whence the isomorphic class of (Θ) ⊗ R P belongs to Pic((Θ)). We have then constructed an element [ P ]

[φ]

[(Θ) ⊗ R P ] of the group

P((Θ)/ R ), where φ is the obvious map

φ : P −→ (Θ) ⊗ R P

( p −→ e ⊗ R p ),

where e ∈ Unit{ p }. It is clear that, for every x ∈ G , the equality φ( P )Θx = Θx φ( P ) holds true in

(Θ) ⊗ R P . Thus, [ P ]

[φ]

[(Θ) ⊗ R P ] ∈ PZ ((Θ)/ R )(G ) and

 S2 [ P ]

[φ]



(Θ) ⊗ R P



= [ P ].

This shows the inclusion Ker(S3 ) ⊆ Im(S2 ).

[Y ] ∈ PZ ((Θ)/ R )G . We need to compute the image S3 ([ Q ]). From the choice of [ Q ] we know that [ Q ] ∈ PicZ ( R ) and that, for every x ∈ G , ψ( Q )Θx = Θx ψ( Q ) in the (Θ)-bimodule Y . As in the proof of Lemma 4.1, the Conversely, let [ Q ] ∈ Im(S2 ), that is, there exists an element [ Q ]

[ψ]

later means that, there are R-bilinear isomorphisms: ∼ =

f x : Q ⊗ R Θx − −→ Θx ⊗ R Q ,

for every x ∈ G ,

L. El Kaoutit, J. Gómez-Torrecillas / Journal of Algebra 370 (2012) 266–296

289

which convert commutative the following diagrams Q ⊗ FxΘ, y

Q ⊗ R Θx ⊗ R Θ y

Q ⊗ R Θxy

f x ⊗Θ y

Y.

f xy

Θx ⊗ R Q ⊗ R Θ y

Θxy ⊗ R Q Θx ⊗ f y

FxΘ, y ⊗ Q

Θx ⊗ R Θ y ⊗ R Q

Coming back to the image S3 ([ Q ]) := [(Γ )], we know that for every x ∈ G , we have Γx = Q ⊗ R Θx ⊗ R Q −1 ∼ = Θx , via the f x ’s. The above commutative diagram shows that these isomorphisms are compatible with the factor maps FxΓ, y and FxΘ, y . Therefore, (Θ) ∼ = (Γ ) as generalized crossed products, and so S3 ([ Q ]) = [(Θ)] the neutral element of the group C0 (Θ/ R ). 2 4.3. Comparison with a previous alternative exact sequence σ

Let us assume here that Θ : G −→ Aut( R ) → Pic( R ), as in Example 2.3. We consider the extension R ⊂ R G , where S := R G = (Θ) is the skew group ring of R by G . Thus Θ : G → Inv R ( S ) factors though out μ as in diagram (17). In what follows, we want to explain the relation between the five terms exact sequence given in [4, Theorem 2.8], and the resulting one by combining Proposition 4.5 and Corollary 4.3, that is the sequence









1 HΘ G , U (Z )

1 S3

S1

 G PZ (Θ)/ R

S2

PicZ ( R )G

2 HΘ G , U (Z ) ∼ = C0 (Θ/ R ).

∗ (G , U (Z )) coincides with that considered in [4]. As we argued in Example 2.3, the cohomology H Θ Thus, following the notation of [4], the result [4, Theorem 2.8] says that

1





ϕσ

1 HΘ G , U (Z )

Fσ

G -Pic( R )

Φσ

Pic( R )G





2 HΘ G , U (Z )

is an exact sequence (of pointed sets). It is not difficult to see that the restriction of Φσ to PicZ ( R )G gives exactly S3 . On the other hand, for any element [ P ]

[ X ] ∈ PZ ((Θ)/ R )G , we will

[φ]

define an object ( P , τ ) where τ : G → AutZ ( P ) (to the group of additive automorphisms of P ) is a G -homomorphism [4, Definition 2.3]. As in the proof of Lemma 4.1, we know that there is a family of R-bilinear isomorphisms jx : R x ⊗ R P → P ⊗ R R x , x ∈ G , so we define τ by the following composition −

τx : x 1 P

∼ =

R x ⊗R P

jx

P ⊗R R x

∼ =

P x,

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L. El Kaoutit, J. Gómez-Torrecillas / Journal of Algebra 370 (2012) 266–296

here P y denotes the R-bimodule constructed from P by twisting its right module structure using the automorphism y, and the same construction is used on the left. To check that τ is a homomorphism of groups, one uses the equality







 

 



jxy ◦ FxΘ, y ⊗ R P = P ⊗ R FxΘ, y ◦ jx ⊗ R R y ◦ R x ⊗ R j y ,

for all x, y ∈ G .

Now, if we take another representative, that is, if we assume that

[P ]

  [X] = P

[φ]

 

[φ ]

X

 G ∈ PZ (Θ)/ R ,

then, by the definition of the group P((Θ)/ R ), see [9, Section 3], there are bilinear isomorphisms f : P → P and g : X → X such that

P

φ

X g

f

P

φ

X

commutes. In this way we have also the following commutative diagram

R x ⊗R P

jx

P ⊗R R x

Rx⊗ f

f ⊗ Rx

R x ⊗R P

j x

P ⊗R R x,

which says that τx ◦ f = f ◦ τx . This implies that [ P , τ ] = [ P , τ ] in the group G -Pic( R ). We then have constructed a monomorphism of groups PZ ((Θ)/ R )G → G -Pic( R ) such that the restriction of the map F σ to PZ ((Θ)/ R )G is exactly S2 . We leave to the reader to check that we furthermore obtain a commutative diagram

1

1 HΘ (G , U (Z ))

1

1 HΘ (G , U (Z ))

S1

PZ ((Θ)/ R )G

ϕσ

G -Pic( R )

S2

Fσ

PicZ ( R )G

Pic( R )G

S3

Φσ

2 HΘ (G , U (Z ))

2 HΘ (G , U (Z )).

4.4. The third exact sequence We define the group B(Θ/ R ) as the quotient group

PicZ ( R )(G )

L

C(Θ/ R )

where L is the map defined in Lemma 4.4.

B(Θ/ R )

1,

(25)

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291

The third exact sequence of groups is stated in the following Proposition 4.6. Let (Θ) be a generalized crossed product of S with G . Then there is an exact sequence of groups

PicZ ( R )G

S3

C0 (Θ/ R )

S4

B(Θ/ R ),

where S3 and S4 are constructed from the sequence (25) and the commutativity of the diagram stated in Lemma 4.4. Proof. The inclusion Im(S3 ) ⊆ Ker(S4 ) is by construction trivial. Now, let [(Γ )] ∈ C0 (Θ/ R ) such that S4 ([(Γ )]) = 1. By the definition of the group B(Θ/ R ), there exists [ P ] ∈ PicZ ( R )(G ) such that L([ P ]) = [(Γ )]. Therefore, we obtain a chain of R-bilinear isomorphisms:

Θx ∼ = Γx ∼ = P ⊗ R Θ x ⊗ R P −1 ,

for all x ∈ G ,

which means that [ P ] ∈ PicZ ( R )G , and so [(Γ )] = S3 ([ P ]) which completes the proof.

2

Remark 4.7. In the case of a unital commutative Galois extension k ⊆ R with a finite Galois group, see for instance [3, p. 396], and taking the extension R ⊆ S and Θ as in Remark 2.13. The homomorphism S4 coincides, up to the isomorphism of Proposition 3.2, with the homomorphism of groups defined in [3, Theorem A12, p. 406] and where the group B(Θ/ R ) coincides with the Brauer group of the k-Azumaya algebras split by R. 4.5. The fourth exact sequence Consider the following sub-group of the Picard group Pic( R ):





Pic0 ( R ) = [ P ] ∈ Pic( R )  P ∼ R , as bimodules . As we have seen in Proposition 1.3, this is an abelian group. Clearly it inherits the G -module structure of Pic( R ): The action is given by x

[ P ] = [Θx ⊗ R P ⊗ R Θx−1 ],

for every x ∈ G .

Lemma 4.8. Keep the above notations. Then the map

C(Θ/ R )   (Γ )



ζ



Z 1 G , Pic0 ( R )





x −→ [Γx ][Θx−1 ]

defines a homomorphism of groups. Furthermore, there is an exact sequence of groups

1

C0 (Θ/ R )

C(Θ/ R )

ζ





Z 1 G , Pic0 ( R ) .

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Proof. The first claim is immediate from the definitions, as well as the inclusion C0 (Θ/ R ) ⊆ Ker(ζ ). Conversely, given an element [(Γ )] ∈ C(Θ/ R ) such that ζ ([(Γ )]) = 1. This implies that for every x ∈ G , there is an isomorphism of R-bimodule Γx ⊗ R Θx−1 ∼ = R. Hence, Γx ∼ = Θx , for every x ∈ G , which means that [(Γ )] ∈ C0 (Θ/ R ). 2 We define another abelian group, which in the commutative Galois case [13], coincides with the first cohomology group of G with coefficients in Pic0 ( R ). It is defined by the following commutative diagram

PicZ ( R )(G )

Z 1 (G , Pic0 ( R )) L

H 1 (G , Pic0 ( R ))

1

ζ

C(Θ/ R ) whose row is an exact sequence. Our fourth exact sequence is given by the following. Proposition 4.9. Let (Θ) be a generalized crossed product of S with G . Then there is an exact sequence of groups S4

C0 (Θ/ R )

B(Θ/ R )

S5





H 1 G , Pic0 ( R ) .

Proof. First we need to define S5 . We construct it as the map which completes the commutativity of the diagram

C0 (Θ/ R ) S4

PicZ ( R )(G )

L

C(Θ/ R ) ζ

Z 1 (G , Pic0 ( R ))

Lc

B(Θ/ R )

1

S5

H 1 (G , Pic0 ( R ))

1,

whose first row is exact. The fact that Im(S4 ) ⊆ Ker(S5 ) is easily deduced from the previous diagram and Lemma 4.8. Conversely, let Ξ = Lc ([(Γ )]), for some [(Γ )] ∈ C(Θ/ R ), be an element in B(Θ/ R ) (here f c denotes the cokernel map of f ), such that S5 (Ξ ) = S5 ◦ Lc ([(Γ )]) = 1. Then (ζ L)c ◦ ζ ([(Γ )]) = 1, which implies that ζ ([(Γ )]) ∈ ζ (L(PicZ ( R )(G ) )). Therefore, there exists an element [ P ] ∈ PicZ ( R )(G ) such that [(Γ )]L([ P ])−1 ∈ Ker(ζ ) = C0 (Θ/ R ) by Lemma 4.8. Now, we have

    −1      = S4 (Γ ) L P −1 S4 (Γ ) L [ P ]     = Lc (Γ ) Lc L P −1   = Lc (Γ ) = Ξ, and this shows that Ξ ∈ Im(S4 ).

2

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293

4.6. The fifth exact sequence and the main theorem Keep from Section 4.5 the definition of the group Pic0 ( R ) with its structure of G -module. Before stating the fifth sequence, we will need first to give a homomorphism of group from the group 3 H 1 (G , Pic0 ( R )) to the third cohomology group H Θ (G , U (Z )). Given a normalized 1-cocycle g ∈ Z 1 (G , Pic0 ( R )) and put g x = [∇x ]. Then, for every pair of elements x, y ∈ G , one can easily shows that x

g y [Θx ] = [Θx ] g y .

(26)

Now, for every x ∈ G , we set

[U x ] := g x [Θx ],

in Pic( R ).

By the cocycle condition, we obtain

[U xy ] = g xy [Θxy ] = g x x g y [Θx ][Θ y ] (26)

= g x [Θx ] g y [Θ y ]

= [U x ][U y ]. This means that there are R-bilinear isomorphisms g

Fx, y : U x ⊗ R U y −→ U xy , g

(27)

g

3 with F1,x = id = Fx,1 for every x ∈ G . By Proposition 2.5, we have a 3-cocycle in Z Θ (G , U (Z )) atg g tached to these maps F−,− , which we denote by α−,−,− . If there is another class [ V x ] ∈ Pic( R ) such that [ V x ] = g x [Θx ], for every x ∈ G , then the families of invertible R-bimodules { V x }x∈G and {U x }x∈G satisfy the conditions of Proposition 2.8. Therefore, the associated 3-cocycles are cohomologous. This g 3 (G , U (Z )) is a well defined map. Henceforth, we means that the correspondence g → [α−,−,− ] ∈ H Θ have a well defined homomorphism of groups





Z 1 G , Pic0 ( R ) g

S13





3 HΘ G , U (Z )





(28)

g . α−,−,−

The proof of the fact that S13 is a multiplicative map uses Lemma 1.1(ii) and the twisted natural transformation of Proposition 1.3. Complete and detailed steps of this proof are omitted. Proposition 4.10. Let R ⊆ S be an extension of rings with the same set of local units, and (Θ) a generalized crossed product of S with G . Then the homomorphism of formula (28) satisfies S13 ◦ ζ = [1], where ζ is the homomorphism of Lemma 4.8. Furthermore, there is a commutative diagram of groups

294

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PicZ ( R )(G )

(ζ ◦L)

Z 1 (G , Pic0 ( R ))

H 1 (G , Pic0 ( R ))

S13

1

S6

3 (G , U (Z )) HΘ

with exact row. Proof. We need to check that S13 ◦ ζ ([(Γ )]) = [1], for any element [(Γ )] ∈ C(Θ/ R ). Set f = f

ζ ([(Γ )]), so we have f x [Θx ] = [Γx ], for every x ∈ G . Thus, the corresponding maps F−.− of forΓ of (Γ ) relative to Γ . They define an associative mula (27), are exactly given by the factor maps F−,− multiplication on (Γ ), and so induce a trivial 3-cocycle, which means that S13 ( f ) = [1]. The last statement is now clear. 2 At this level we are ending up with the following commutative diagram

PicZ ( R )G

S3

C0 (Θ/ R ) S4

1

PicZ ( R )(G )

L

Lc

C(Θ/ R )

B(Θ/ R ) S5

ζ

PicZ ( R )(G )

(ζ ◦L)

1

(ζ ◦L)

c

Z 1 (G , Pic0 ( R )) S13

H 1 (G , Pic0 ( R ))

1.

S6

3 HΘ (G , U (Z ))

Proposition 4.11. Let (Θ) be a generalized crossed product of S with G . Then there is an exact sequence of groups

B(Θ/ R )

S5





H 1 G , Pic0 ( R )

S6





3 HΘ G , U (Z ) .

Proof. It is clear from the above diagram that Im(S5 ) ⊆ Ker(S6 ) since by Proposition 4.10 S13 ◦ ζ = [1]. Conversely, let us consider a class [h] ∈ H 1 (G , Pic0 ( R )) such that S6 ([h]) = 1, for some element h ∈ 3 Z 1 (G , Pic0 ( R )). By the same diagram, we also have that the class S13 (h) = [1] ∈ H Θ (G , U (Z )). This 3 says that S13 (h) = [β], for some β ∈ B Θ (G , U (Z )), and so there exists σ−,− : G × G → U (Z ) such that −1 βx, y ,z = σx, yz x σ y ,z σx−, y1 σxy ,z , h for every x, y , z ∈ G . Now, for each x ∈ G , we put h x [Θx ] = [Ωx ], and consider F−,− the associated

family of maps as in (27), Fxh, y : Ωx ⊗ R Ω y → Ωxy , where x, y ∈ G . By definition the β−,−,− ’s satisfy

    h h βx, y ,z ◦ Fxh, yz ◦ Fxh, y ⊗ R Ωz = Fxy , z ◦ Ωx ⊗ R F y , z , Ω relative to Ω , defined by for every x, y , z ∈ G . In this way, there are factor maps F−,−

L. El Kaoutit, J. Gómez-Torrecillas / Journal of Algebra 370 (2012) 266–296

FxΩ, y : Ωx ⊗ R Ω y

Ωxy

ωx ⊗ R ω y

σx, y (e)Fxh, y (ωx ⊗ R ω y ),

295

where e ∈ Unit{ωx , ω y }. A routine computation shows that (Ω) is actually a generalized crossed Ω relative to Ω . product of S with G with factor maps F−,− On the other hand, since for each x ∈ G , h x ∈ Pic0 ( R ), we have Ωx ∼ Θx . Thus [(Ω)] ∈ C(Θ/ R ) with ζ ([(Ω)]) = h. Whence

     (ζ L)c (h) = [h] = (ζ L)c ◦ ζ (Ω) = S5 Lc (Ω) , which implies that [h] ∈ Im(S5 ), and this completes the proof.

2

The following diagram summarizes the information we have shown so far.

1 HΘ (G , U (Z ))

S1

S2

PZ (/ R )

PicZ ( R )G

S3

2 HΘ (G , U (Z ))

S4 L

PicZ ( R )(G )

Lc

C(Θ/ R )

B(Θ/ R ) S5

ζ

PicZ ( R )(G )

(ζ ◦L)

Z 1 (G , Pic0 ( R ))

(ζ ◦L)c

S13

H 1 (G , Pic0 ( R )).

S6

3 HΘ (G , U (Z ))

We are now in position to announce our main result. Theorem 4.12. Let R ⊆ S be an extension of rings with the same set of local units, and (Θ) a generalized crossed product of S with a group G . Then there is an exact sequence of groups





1 HΘ G , U (Z )

1 S4

B(Θ/ R )

S5

S1

PZ (/ R )G 



H 1 G , Pic0 ( R )

S2

S6

S3

PicZ ( R )G







2 HΘ G , U (Z )



3 HΘ G , U (Z ) .

Proof. This is a direct consequence of Corollary 4.3 and Propositions 4.5, 4.6, 4.9, and 4.11.

2

Acknowledgment We thank the referee for a careful reading of our manuscript and for pointing out the existence of the paper [4]. References [1] G.D. Abrams, Morita equivalence for rings with local units, Comm. Algebra 11 (8) (1983) 801–837. [2] P.N. Ánh, L. Márki, Rees matrix rings, J. Algebra 81 (1983) 340–369. [3] M. Auslander, O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960) 367–409.

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